Wigner functions of the finite square-well bound states Free

The finite square-well potential is commonly taught in introductory quantum mechanics, but its eigen-energies are determined by transcendental equations. These have been solved numerically or graphically in a number of papers.^1–7 In addition to the eigen-energies, the bound states and their non-classical properties are of importance to simulate and understand the quantum dynamics of a particle confined in a finite-well potential.⁸ 

Here, we focus on understanding the non-classicality of the bound states of the finite well using their Wigner functions.⁹ Wigner functions are useful for simulating the wave-package evolution of a particle in a coherent state and monitoring its non-classicality over time.⁸ Here, we use the Wigner function to study the non-classicality of the stationary states. In the emerging field of quantum technologies, the non-classicality itself has been regarded as a useful resource (for example, its application in quantum metrology¹⁰) and the Wigner function has been widely adopted for quantum information tasks, such as to measure and reconstruct an unknown quantum state via so-called quantum tomography.¹¹ 

Dividing Eqs. (3) and (4) produces the transcendental equation $η = ξ tan ξ$⁠. Inserting these wave functions into the time-independent Schrödinger equation produces $ξ 2 + η 2 = R 2$⁠, where $R = a m V 0 / ( 2 ℏ 2 )$⁠. (Therefore, at the infinite-well limit, i.e., $V 0 → ∞ , R → ∞$⁠.) These two equations can be used to find numerical solutions of $( ξ , η )$⁠. These give a set of solutions, indicated hereinafter by $( ξ n , η n )$ with n = 1, 3, $⋯$ for the even-parity case and n = 2, 4, $…$ for the odd-parity case. For a given R, the total number of solutions is given by $N = Ceiling ( 2 R / π )$⁠, where $Ceiling ( x )$ gives the smallest integer greater than or equal to x. With the relation $k n = 2 ξ n / a$⁠, one can obtain the eigen-energies $E n = ( ξ n / R ) 2 V 0$ (see Ref. 14). Using $( ξ n , η n )$⁠, one can also obtain the coefficients of the wave functions $A ̃$ and $B ̃$⁠, which in turn give A and B, as well as the bound states $ψ n ( x )$⁠.

This simple relationship between the Wigner function and probability densities in both position and momentum space makes the Wigner function very much like a probability density, with one exception. The Wigner function can take negative values, but a negative probability has no meaning. The answer to this apparent puzzle lies in the uncertainty principle. The Wigner function may be negative at some given (x, p), but the uncertainty principle means it is not possible to make simultaneous exact measurements of x and p. Negativity in Wigner functions is always balanced by larger positive values within $ℏ$-sized volumes of phase space so that the net probability is positive. However, when there is significant negativity in the Wigner function, the net probability depends very much on how that volume is shaped, indicating that the particle behaves in a non-classical way near this region of phase space.

According to Ref. 18, the degree of non-classicality can be quantified by $v = 2 I − / ( I + + I − )$⁠, where $I +$ is the integral of the Wigner function over those domains where it is positive (i.e., in positive-valued regions) and $I −$ is the absolute value of the integral of the Wigner function over negative-valued regions. Using the Wigner function $W n ( x , p )$ and introducing $k ̃ ≡ p / ℏ$ (in units of $1 / a$⁠), one can easily obtain $∫ − ∞ ∞ ∫ − ∞ ∞ W n ( x , k ̃ ) dxd k ̃ = I + − I − = 2$⁠. One may define the negative fraction $v = I − / ( I − + 1 )$⁠, with $v ∈ [ 0 , 1 )$⁠. For all classical states (e.g., the coherent state), one finds v = 0 (see Ref. 18). However, for a non-classical state, v is nonvanishing. The bigger v is, the more non-classical is the state.¹⁸ 

Note that it is quite hard to calculate $I −$ (and hence v) since its calculation requires a double integral over an infinite phase space (x, p) or equivalently (⁠ $x , k ̃$⁠). To roughly evaluate the degree of the non-classicality, here we calculate $I −$ within the local region $x ∈ [ − a , a ]$ and $k ̃ ∈ [ − 20 / a , 20 / a ]$⁠, as depicted by Fig. 1.

Figure 1 shows density plots of the Wigner functions for the infinite-well case (the left panels) and the finite-well case (the right panels), by taking R = 5 (i.e., N = 4) and the number of points sampled in each dimension N_s = 160. For simplicity, we set a = 1, and therefore, the position is in units of a. The dotted contour lines enclose the positive regions of the Wigner function and the negative contour lines enclose the negative regions with $W n ( x , p ) = 0.1$ (⁠ $− 10 − 8$⁠). For the finite-well case, we calculate $I −$ with respect to the right panel of Fig. 1 and obtain the degree of non-classicality $v = I − / ( I − + 1 ) ≈ 4.03 × 10 − 2$⁠, 0.321, and 0.535 (from top to bottom). These results indicate that the bound states become more and more non-classical with increasing of n.

Next, we investigate the details of the Wigner functions. For the Wigner functions for the infinite-well case,¹⁹ the Wigner functions are vanishing outside the well, as expected. Furthermore, the Wigner functions of $ψ n ( ∞ )$ are most concentrated in the central region, i.e., $x ∈ ( − a / 2 , a / 2 )$ and $k ̃ = p / ℏ ∈ ( − n π / a , n π / a )$⁠, with n = 1, 2, 3, $…$⁠. For the ground state, the value of $W 1 ( x , p )$ is always positive in the central region, enclosed by the dotted contour line of Fig. 1(a). For highly excited states, the values of $W n ( x , p )$ change rapidly from positive to negative, exhibiting an interference pattern.

The Wigner functions are symmetric in phase space (x, p) for both the finite and infinite wells, since the Schrödinger equation is invariant under translations $x → − x$ and $p → − p$ and the potential is symmetric. By comparing Figs. 1(a), 1(c), and 1(e) with Figs. 1(b), 1(d), and 1(f), one can see that the finite-well Wigner functions extend outside the well and the interference pattern is more complicated. Furthermore, the Wigner functions for each ψ_n get squeezed along the momentum p. One can easily see this by comparing the central region enclosed by the dotted contour lines for the finite well states to the infinite well states. This can be understood by the following argument: Since the wave function is not exactly zero outside the finite well, the finite well states have a longer wavelength for any given value of n, which means a smaller momentum distribution of ψ_n. For the infinite-well case, the wave functions go to zero at the edges of the well, so the Fourier transform of $ψ n ( ∞ )$ requires a broader momentum spectrum.

The number of maxima of $W n ( x , 0 )$ increases with n, similar to that of $| ψ n ( x ) | 2$ (see Ref. 14 for more discussion). The color scale of Fig. 1(b) was chosen so that the behavior of the Wigner functions near the central peak is most visible. On this scale, the Wigner function of ground state for the finite well in Fig. 1(b) appears to take only positive values. However, for this ground state, the values of $W 1 ( x , p )$ alternate between positive and negative in the fringes outside of the central region. To see this clearly, we choose three points inside the well [with their locations indicated by A, B, and C of Fig. 1(b)] and find $W 1 = − 1.41 × 10 − 2 , 4.69 × 10 − 3$⁠, and $− 1.89 × 10 − 3$⁠, respectively. Outside the well, for $| x | > a / 2$⁠, this phenomenon also occurs. Figure 1(d) shows the Wigner function of the first excited state (i.e., n = 2) with points A, B, and C chosen to be outside the well. The values of $W 2 ( x , p )$ at the three points of Fig. 1(d) also alternate sign in the successive fringes, e.g., $W 2 = − 1.11 × 10 − 3 , 7.60 × 10 − 5$⁠, and $− 1.44 × 10 − 5$⁠. With the increase in n, the negative-value region becomes larger and larger, which indicates that the bound states show more non-classicality for high n. As depicted by Fig. 1(f), within the central region, there are two negative-value regions of the Wigner function for the highest excited states (i.e., N = 4 for R = 5), with the points A and B inside. The value of $W N ( x , p )$ at the three points are all negative, with $W N = − 0.637$⁠, –0.265, and $− 5.32 × 10 − 2$⁠.

In summary, we have investigated the bound states and their non-classical properties for a particle confined in a one-dimensional finite square well, using the Wigner functions for the states in the phase space (x, p). Our numerical results demonstrated that the Wigner functions show a more complicated structure than the infinite-well case: (i) the central peak spreads over the x axis and gets squeezed along the momentum p axis compared to the infinite well states for each quantum level n; (ii) the values of $W n ( x , p )$ alternate between negative and positive in the successive fringes; and (iii) the magnitude of negativity of the bound states increases with n, evidence of more and more non-classicality. These observations are useful in class to distinguish ψ_n and $ψ n ( ∞ )$⁠.

To the best of his knowledge, the author has no conflict of interest to disclose.

This research was funded by the Science Foundation of Zhejiang Sci-Tech University (Grant No. 18062145-Y) and partially by the National Natural Science Foundation of China (NSFC) (Grant No. 12075209).